Most Irish do not self-identify as "British," but I believe that the average Scot or Welshman readily accepts that designation (although, of course, they would balk at being described as "English").

Scotland and Wales, of course, share the same island with England. Perhaps that helps them feel British.

Also, while the Celtic peoples of those two nations endured conquest, and a degree of discrimination, by England, their subjugation was hardly as severe nor as long-standing as that suffered by Ireland.